Mathematics thesis topics
We offer a vibrant and supportive environment for postgraduate research in Mathematics. Our commitment to innovative and impactful research fosters the development of ideas that contribute significantly to various fields, from theoretical investigations to practical applications.
Research Groups and Collaborations
Our department hosts various research groups that foster collaboration and interdisciplinary work. Collaborations with industry partners and other academic institutions also enhance the relevance and application of our research. Visit Research in Mathematics and Research in Statistics and Data Analytics to find out more about our research groups.
Sample thesis topics
Below, you can find a selection of sample thesis topics available for prospective postgraduate research students. These topics illustrate the wide range of research areas we offer, but they are not exhaustive. Most prospective supervisors welcome discussions about potential projects beyond those listed here. Additionally, funded projects come with specific funding attached, while funding for other projects is typically available on a competitive basis.
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More information: Research
Centres of Doctoral Training (CDTs)
The EPSRC Centre for Doctoral Training in Algebra, Geometry, and Quantum Fields (AGQ CDT)
The AGQ CDT is a collaborative effort by University of Glasgow, University of Edinburgh, and Heriot-Watt University. The programme is at the forefront of mathematical innovation, leveraging the power of symmetry, geometry, and quantum physics to shape the technologies of tomorrow. Application information can be found on their How to Apply page.
The Leverhulme Programme for Doctoral Training in Ecological Data Science
Our Leverhulme Programme for Doctoral Training in Ecological Data Science will train a new generation of data scientists. Students will be equipped with the skills to tackle the most pressing environmental challenges of our time and be trained in the latest data science techniques. Application information can be found on their Apply page.
DiveIn (EPSRC CDT in Diversity-Led, Mission-Driven Research)
The DiveIn CDT prioritises diversity, creating an inclusive space for varied talents to produce transformative interdisciplinary research in Net Zero, AI and Big Data, Technology Touching Life, Future Telecoms, Quantum Technologies and more. Application information can be found on their Apply page.
ExaGEO
ExaGEO (Exascale computing for Earth, Environmental, and Sustainability Solutions) train the next generation of Earth and environmental scientists to harness the power of exascale computing. The following projects are currently available via ExaGEO, and details can be found on the website about how apply.
Detecting hotspots of water pollution in complex constrained domains and networks (PhD)
Supervisors: Mu Niu, Craig Wilkie, Cathy Yi-Hsuan Chen (Business School, UofG), Michael Tso (Lancaster University)
Relevant research groups: Modelling in Space and Time, Bayesian Modelling and Inference, Environmental, Ecological Sciences & Sustainability
Technological developments with smart sensors are changing the way that the environment is monitored. Many such smart systems are under development, with small, energy efficient, mobile sensors being trialled. Such systems offer opportunities to change how we monitor the environment, but this requires additional statistical development in the optimisation of the location of the sensors.
The aim of this project is to develop a mathematical and computational inferential framework to identify optimal sensor deployment locations within complex, constrained domains and networks for improved water contamination detection. Methods for estimating covariance functions in such domains rely on computationally intensive diffusion process simulations, limiting their application to relatively simple domains and small-scale datasets. To address this challenge, the project will employ accelerated computing paradigms with highly parallelized GPUs to enhance simulation efficiency. The framework will also address regression, classification, and optimization problems on latent manifolds embedded in high-dimensional spaces, such as image clouds (e.g., remote sensing satellite images), which are crucial for sensor deployment and performance evaluation. As the project progresses, particularly in the image cloud case, the computational demands will intensify, requiring advanced GPU resources or exascale computing to ensure scalability, efficiency, and performance.
Developing GPU-accelerated digital twins of ecological systems for population monitoring and scenario analyses (PhD)
Supervisors: Colin Torney, Juan Morales (BOHVM, UoG), Rachel McCrea (Lancaster University), Tiffany Vlaar, Dirk Husmeier
Relevant research groups: Machine Learning and AI, Emulation and Uncertainty Quantification, Environmental, Ecological Sciences & Sustainability, Mathematical Biology
This PhD project focuses on advancing ecological research by using high-resolution datasets and GPU computing to develop digital twins of ecological systems. The study will concentrate on a population of free-roaming sheep in Patagonia, Argentina, examining the relationship between individual decision-making and population dynamics. Using data from state-of-the-art GPS collars, the research will investigate the impact of an individual’s condition on activity budgets and space use, and the dual influence of parasites on behaviour and energy balance. The digital twins will enhance the accuracy of population-level predictions and offer a versatile and transferable framework for ecosystem monitoring, providing critical insights for environmental policy, conservation strategies, and sustainable food systems.
Downscaling and Prediction of Rainfall Extremes from Climate Model Outputs (PhD)
Supervisors: Sebastian Gerhard Mutz (GES, UoG), Daniela Castro-Camilo
Relevant research groups: Modelling in Space and Time, Bayesian Modelling and Inference, Environmental, Ecological Sciences & Sustainability
In the last decade, Scotland’s rainfall increased by 9% annually and 19% in winter, with more water from extreme events, posing risks to the environment, infrastructure, health, and industry. Urgent issues such as flooding, mass wasting, and water quality are closely tied to rainfall extremes. Reliable predictions of extremes are, therefore, critical for risk management. Prediction of extremes, which is one of the main focuses of extreme value theory, is still considered one of the grand challenges by the World Climate Research Programme. This project will address this challenge by developing novel statistical, computationally efficient models that are able to predict rainfall extremes from the output of GPU-optimised climate models.
Exploring Hybrid Flood modelling leveraging GPU/Exascale computing (PhD)
Supervisors: Andrew Elliott, Lindsay Beevers (University of Edinburgh), Claire Miller, Michele Weiland (University of Edinburgh)
Relevant research groups: Modelling in Space and Time, Environmental, Ecological Sciences & Sustainability, Machine Learning and AI, Emulation and Uncertainty Quantification
Flood modelling is crucial for understanding flood hazards, now and in the future as a result of climate change. Modelling provides inundation extents (or flood footprints) which provide outlines of areas at risk which can help to manage our increasingly complex infrastructure network as our climate changes. Our ability to make fast, accurate predictions of fluvial inundation extents is important for disaster risk reduction. Simultaneously capturing uncertainty in forecasts or predictions is essential for efficient planning and design. Both aims require methods which are computationally efficient whilst maintaining accurate predictions. Current Navier-stokes physics-based models are computationally intensive; thus this project would explore approaches to hybrid flood models which utilise GPU-compute and ML fused with physics-based models, as well as investigating scaling the numerical models to large-scale HPC resources.
Scalable approaches to mathematical modelling and uncertainty quantification in heterogeneous peatlands (PhD)
Supervisors: Raimondo Penta, Vinny Davies, Jessica Davies (Lancaster University), Lawrence Bull, Matteo Icardi (University of Nottingham)
Relevant research groups: Modelling in Space and Time, Environmental, Ecological Sciences & Sustainability, Machine Learning and AI, Emulation and Uncertainty Quantification, Continuum Mechanics
While only covering 3% of the Earth’s surface, peatlands store >30% of terrestrial carbon and play a vital ecological role. Peatlands are, however, highly sensitive to climate change and human pressures, and therefore understanding and restoring them is crucial for climate action. Multiscale mathematical models can represent the complex microstructures and interactions that control peatland dynamics but are limited by their computational demands. GPU and Exascale computing advances offer a timely opportunity to unlock the potential benefits of mathematically-led peatland modelling approaches. By scaling these complex models to run on new architectures or by directly incorporating mathematical constraints into GPU-based deep learning approaches, scalable computing will to deliver transformative insights into peatland dynamics and their restoration, supporting global climate efforts.
Scalable Inference and Uncertainty Quantification for Ecosystem Modelling (PhD)
Supervisors: Vinny Davies, Richard Reeve (BOHVM, UoG), David Johnson (Lancaster University), Christina Cobbold, Neil Brummitt (Natural History Museum)
Relevant research groups: Modelling in Space and Time, Environmental, Ecological Sciences & Sustainability, Machine Learning and AI, Emulation and Uncertainty Quantification
Understanding the stability of ecosystems and how they are impacted by climate and land use change can allow us to identify sites where biodiversity loss will occur and help to direct policymakers in mitigation efforts. Our current digital twin of plant biodiversity – https://github.com/EcoJulia/EcoSISTEM.jl – provides functionality for simulating species through processes of competition, reproduction, dispersal and death, as well as environmental changes in climate and habitat, but it would benefit from enhancement in several areas. The three this project would most likely target are the introduction of a soil layer (and the improvement of the modelling of soil water); improving the efficiency of the code to handle a more complex model and to allow stochastic and systematic Uncertainty Quantification (UQ); and developing techniques for scalable inference of missing parameters.
Smart-sensing for systems-level water quality monitoring (PhD)
Supervisors: Craig Wilkie, Lawrence Bull, Claire Miller, Stephen Thackeray (Lancaster University)
Relevant research groups: Machine Learning and AI, Emulation and Uncertainty Quantification, Environmental, Ecological Sciences & Sustainability
Freshwater systems are vital for sustaining the environment, agriculture, and urban development, yet in the UK, only 33% of rivers and canals meet ‘good ecological status’ (JNCC, 2024). Water monitoring is essential to mitigate the damage caused by pollutants (from agriculture, urban settlements, or waste treatment) and while sensors are increasingly affordable, coverage remains a significant issue. New techniques for edge processing and remote power offer one solution, providing alternative sources of telemetry data. However, methods which combine such information into systems-level sensing for water are not as mature as other applications (e.g., built environment). In response, procedures for computation at the edge, decision-making, and data/model interoperability are considerations of this project.
Statistical Emulation Development for Landscape Evolution Models (PhD)
Supervisors: Benn Macdonald, Mu Niu, Paul Eizenhöfer (GES, UoG), Eky Febrianto (Engineering, UoG)
Relevant research groups: Modelling in Space and Time, Environmental, Ecological Sciences & Sustainability, Machine Learning and AI, Emulation and Uncertainty Quantification
Many real-world processes, including those governing landscape evolution, can be effectively mathematically described via differential equations. These equations describe how processes, e.g. the physiography of mountainous landscapes, change with respect to other variables, e.g. time and space. Conventional approaches for performing statistical inference involve repeated numerical solving of the equations. Every time parameters of the equations are changed in a statistical optimisation or sampling procedure; the equations need to be re-solved numerically. The associated large computational cost limits advancements when scaling to more complex systems, the application of statistical inference and machine learning approaches, as well as the implementation of more holistic approaches to Earth System science. This yields to the need for an accelerated computing paradigm involving highly parallelised GPUs for the evaluation of the forward problem.
Beyond advanced computing hardware, emulation is becoming a more popular way to tackle this issue. The idea is that first the differential equations are solved as many times as possible and then the output is interpolated using statistical techniques. Then, when inference is carried out, the emulator predictions replace the differential equation solutions. Since prediction from an emulator is very fast, this avoids the computational bottleneck. If the emulator is a good representation of the differential equation output, then parameter inference can be accurate.
The student will begin by working on parallelising the numerical solver of the mathematical model via GPUs. This means that many more solutions can be generated on which to build the emulator, in a timeframe that is feasible. Then, they will develop efficient emulators for complex landscape evolution models, as the PhD project evolves.
Towards exa-scale simulations of slabs, core-mantle heterogeneities and the geodynamo (PhD)
Supervisors: Radostin Simitev, Antoniette Greta Grima (GES, UoG), Dr Kevin Stratford (University of Edinburgh)
Relevant research groups: Geophysical & astrophysical fluid dynamics
Scientific computing is crucial for understanding geophysical fluid flows, such as the geodynamo that sustains Earth’s magnetic field. This project will adapt an existing pseudo-spectral geodynamo code for magnetohydrodynamic simulations in rotating spherical geometries to GPU architectures, improving efficiency on modern computing systems and enabling simulations of more realistic regimes. This will advance our understanding of Earth’s geomagnetic field and its broader interactions, such as those with mantle heterogeneities. Evidence from seismology and geodynamics shows that the core-mantle boundary (CMB) is highly heterogeneous, influencing heat transport and geodynamo dynamics. By combining compressible, thermochemical convection with geodynamo simulations, this project will further investigate how deep slab properties affect the CMB heat flux, mantle heterogeneity, and the geodynamo.
IAPETUS2
Named after the ancient ocean that closed to bring together England and Scotland. Iapetus2 is a partnership that joins the leading research universities of Durham, Heriot Watt, Glasgow, Newcastle, St Andrews and Stirling, together with the British Antarctic Survey, British Geological Survey and the Centre for Ecology & Hydrology, in a united approach to doctoral research and training the next generation of leaders in the science of the natural environment. Application information can be found on their Apply page.
NETGAIN
NETGAIN (developing the science and practice of nature markets for a net positive future) is a collaborative CDT held between the Universities of St Andrews, Aberdeen, Durham and Glasgow. NETGAIN will train a new generation of multidisciplinary scientist-practitioners to transform the landscape of nature markets, ensuring effective, evidence-based solutions to the world’s most urgent environmental challenges. The application process and potential project will be linked from here soon.
Pure Mathematics
Algebra - Example Research Projects
Our group has an active PhD student community, and every year we admit new PhD students. We welcome applications from across the world, and we encourage you to browse our available supervisors, and also to consult our general advice on how to navigate the application process.
There is also the possibility of applying to The EPSRC Centre for Doctoral Training in Algebra, Geometry, and Quantum Fields (AGQ CDT) is a collaborative effort by University of Glasgow, University of Edinburgh, and Heriot-Watt University. The programme is at the forefront of mathematical innovation, leveraging the power of symmetry, geometry, and quantum physics to shape the technologies of tomorrow.
Algebraic Geometry - Example Research Projects
Our group has an active PhD student community, and every year we admit new PhD students. We welcome applications from across the world, and we encourage you to browse our available supervisors, and also to consult our general advice on how to navigate the application process.
There is also the possibility of applying to The EPSRC Centre for Doctoral Training in Algebra, Geometry, and Quantum Fields (AGQ CDT) is a collaborative effort by University of Glasgow, University of Edinburgh, and Heriot-Watt University. The programme is at the forefront of mathematical innovation, leveraging the power of symmetry, geometry, and quantum physics to shape the technologies of tomorrow.
Analysis - Example Research Projects
Our group has an active PhD student community, and every year we admit new PhD students. We welcome applications from across the world, and we encourage you to browse our available supervisors, and also to consult our general advice on how to navigate the application process.
There is also the possibility of applying to The EPSRC Centre for Doctoral Training in Algebra, Geometry, and Quantum Fields (AGQ CDT) is a collaborative effort by University of Glasgow, University of Edinburgh, and Heriot-Watt University. The programme is at the forefront of mathematical innovation, leveraging the power of symmetry, geometry, and quantum physics to shape the technologies of tomorrow.
Geometry and Topology - Example Research Projects
Our group has an active PhD student community, and every year we admit new PhD students. We welcome applications from across the world, and we encourage you to browse our available supervisors, and also to consult our general advice on how to navigate the application process.
There is also the possibility of applying to The EPSRC Centre for Doctoral Training in Algebra, Geometry, and Quantum Fields (AGQ CDT) is a collaborative effort by University of Glasgow, University of Edinburgh, and Heriot-Watt University. The programme is at the forefront of mathematical innovation, leveraging the power of symmetry, geometry, and quantum physics to shape the technologies of tomorrow.
Number Theory - Example Research Projects
Our group has an active PhD student community, and every year we admit new PhD students. We welcome applications from across the world, and we encourage you to browse our available supervisors, and also to consult our general advice on how to navigate the application process.
There is also the possibility of applying to The EPSRC Centre for Doctoral Training in Algebra, Geometry, and Quantum Fields (AGQ CDT) is a collaborative effort by University of Glasgow, University of Edinburgh, and Heriot-Watt University. The programme is at the forefront of mathematical innovation, leveraging the power of symmetry, geometry, and quantum physics to shape the technologies of tomorrow.
Applied Mathematics
Continuum Mechanics - Example Research Projects
Information about postgraduate research opportunities and how to apply can be found on the Postgraduate Research Study page. Below is a selection of projects that could be undertaken with our group.
Continuous production of solid metal foams (PhD)
Supervisors: Peter Stewart
Relevant research groups: Continuum Mechanics
Porous metallic solids, or solid metal foams, are exceedingly useful in many engineering applications, as they can be manufactured to be strong yet exceedingly lightweight. However, industrial processing methods for producing such foams are problematic and unreliable, and it is not currently possible to control the porosity distribution of the final product a priori.
This project will consider a new method of solid foam production, where bubbles of gas are introduced continuously into a molten metal flowing through a heat exchanger; foaming and solidification then occur almost simulatanously, allowing the foam structure to be controlled pointwise. The aim of this project is to construct a simple mathematical model for a gas bubble moving in a liquid filled channel ahead of a solidification front, to predict optimal conditions whereby the gas bubble is drawn toward the phase boundary, hence forming a porous solid.
This project will require some background in fluid mechanics and a combination of analytical and numerical techniques for solving partial differential equations.
Radial foam fracture (PhD)
Supervisors: Peter Stewart
Relevant research groups: Continuum Mechanics
Gas-liquid foams are a useful analgoue of crystalline atomic solids. 2D foam fracture has been used to study the mechanisms of fracture in metals. A two-dimenisonal network model (formed from a large system of differential equations) has recently been produced to study foam fracture in a rectangular channel which is pressurised along one edge. This model has helped to explain the origin of the velocity gap (a range of inadmissable steady fracture velocities), observed both in foam fracture experiments and in atomistic simulations of brittle fracture. This project will apply this network modelling approach to study radial foam fracture in a Hele-Shaw cell, to mimick recent experiments. This system has strong similarity to radial Saffmann-Taylor fingering, where fingering has been observed when a less viscous fluid displaces a more viscous fluid in a confined geometry. This project will involve studying systems of ordinary and partial differential equations using both numerical and analytical methods.
Numerical simulations of planetary and stellar dynamos (PhD)
Supervisors: Radostin Simitev
Relevant research groups: Geophysical & Astrophysical Fluid Dynamics, Continuum Mechanics
Using Fluid Dynamics and Magnetohydrodynamics to model the magnetic fields of the Earth, planets, the Sun and stars. Involves high-performance computing.
Mathematical models of vasculogenesis (PhD)
Supervisors: Peter Stewart
Relevant research groups: Mathematical Biology, Continuum Mechanics
Vasculogenesis is the process of forming new blood vessels from endothelial cells, which occurs during embryonic development. Viable blood vessels facilitate tissue perfusion, allowing the tissue to grow beyond the diffusion-limited size. However, in the absence of vasculogenesis, efforts to engineer functional tissues (eg for implantation) are restricted to this diffusion-limited size. This project will investigate mathematical models for vasculogenesis and explore mechanisms to stimulate blood vessel formation for in vitro tissues. The project will involve collaboration with Department of Biological Engineering at MIT, as part of the SofTMechMP project.
A coupled cardiovascular-respiration model for mechanical ventilation (PhD)
Supervisors: Peter Stewart, Nicholas A Hill
Relevant research groups: Mathematical Biology, Continuum Mechanics
Mechanical ventilation is a clinical treatment used to draw air into the lungs to facilitate breathing, used in treatment of premature babies with respiratory distress syndrome and in the treatment of severe Covid pneumonia. The aim is to oxygenate the blood while simultaneously removing unwanted by-products. However, over-inflation of the lungs can reduce the blood supply to the gas exchange surfaces, leading to a ventilation-perfusion mis-match. This PhD project will give you the opportunity to develop a mathematical model to describe the coupling between blood flow in the pulmonary circulation and air flow in the lungs (during both inspiration and expiration). You will devise a coupled computational framework, capable of testing patient-specific ventilation protocols. This is an ideal project for a postgraduate student with an interest in applying mathematical modelling and image analysis to predictive healthcare. The project will give you the opportunity to join a cross-disciplinary Research Hub that aims to push the boundaries of quantitative medicine and improve clinical decision making using innovative mathematical and statistical modelling.
Observationally-constrained 3D convective spherical models of the solar dynamo (Solar MHD) (PhD)
Supervisors: Radostin Simitev, David MacTaggart, Robert Teed
Relevant research groups: Geophysical & Astrophysical Fluid Dynamics, Continuum Mechanics
Solar magnetic fields are produced by a dynamo process in the Solar convection zone by turbulent motions acting against Ohmic dissipation. Solar magnetic activity affects nearEarth space environment and can harm modern technology and endanger human health. Further, Solar magnetism poses fundamental physical and mathematical problems, e.g. about the nature of plasma turbulence and the topology of magnetic field generation. Current models of the global Solar dynamo fall in two classes (a) mean-field dynamos (b) convection-driven dynamos. The mean-field models are only phenomenological as they replace turbulent interactions by ad-hoc source and quenching terms. On the other hand, spherical convection-driven dynamo models are derived from basic principles with minimal assumptions and potentially offer true predictive power; these can also be extended to other stars and giant planets. However, at present, convection driven dynamo models operate in a wrong dynamical regime and have limited success in reproducing a number of important 1 observations including (a) the sunspot cycle period, polarity reversals and the sunspot butterfly diagram, (b) the poleward migration of diffuse surface magnetic fields, (c) the polar field strength and phase relationships between poloidal/toroidal components. The aims of this project are to (a) develop a three-dimensional convection-driven Solar dynamo model constrained by assimilation of helioseismic data, and (b) start to use the model to estimate turbulent properties that determine the internal dynamics and activity cycles of the Sun.
Modelling the force balance in planetary dynamos (PhD)
Supervisors: Robert Teed, Radostin Simitev
Relevant research groups: Geophysical & Astrophysical Fluid Dynamics, Continuum Mechanics
Current simulations of magnetic field (the 'dynamo process') generation in planets are run, not under the conditions of planetary cores and atmospheres, but in a regime idealised for computations. To forecast changes in planetary magnetic fields such as reversals and dynamo collapse, it is vital to understand the actual fluid dynamics of these regions. The aim of this project is to produce simulations of planetary cores and atmospheres with realistic force balances and, in doing so, understand how such force balances arise and affect the dynamics of the flow. The importance of different forces (e.g. Coriolis, Lorentz, viscous forces) determine the dynamics, the dynamo regime, and hence the morphology and strength of the magnetic field that is produced. This project would involve working with existing numerical code to perform the simulations and developing new techniques to determine the heirarchy of forces at play.
Identifying waves in dynamo models (PhD)
Supervisors: Robert Teed
Relevant research groups: Geophysical & Astrophysical Fluid Dynamics, Continuum Mechanics
This project would involve using existing (and developing new) techniques to isolate and study magnetohydrodynamic (MHD) waves in numerical calculations. Various classes of waves exist and may play a role in the dynamo process (which generates planetary magnetic fields) and/or help us better understand changes in the magnetic field.
Magnetic helicity as the key to dynamo bistability (PhD)
Supervisors: David MacTaggart, Radostin Simitev
Relevant research groups: Geophysical & Astrophysical Fluid Dynamics, Continuum Mechanics
The planets in the solar system exhibit very different types of large-scale magnetic field.The Earth has a strongly dipolar field, whereas the fields of other solar system planets, such as Uranus and Neptune, are far more irregular. Although the different physical compositions of the planets of the solar system will influence the behaviour of the large-scale magnetic fields that they generate, the morphology of planetary magnetic fields can depend on properties of dynamos common to all planets. Here, we refer to an important and recent discovery from dynamo simulations. Remarkably, two very different types of chaotic dipolar dynamo solutions have been found to exist over identical values of the basic parameters of a generic model of convection-driven dynamos in rotating spherical shells. The two solutions mentioned above can be characterised as ‘mean dynamos’, MD, where a strong poloidal field dominates and ‘fluctuating dynamos’, FD, where the poloidal component is weaker and the large-scale field can be described as multipolar. Although these two states have been shown to be bistable (co-exist) for a wide range of identical parameters, it is not clear how a particular state, MD or FD, is chosen and how/when one state can change to the other. Some of the bifurcations of such states has been investigated, but a deep understanding of the dynamics that cause the bifurcations remains to be developed. Since the magnetic topology of MD and FD states are fundamentally different, an important part of this project will be to probe the nature of MD and FD states by studying magnetic helicity, a magnetohydrodynamic invariant that combines information on the topology of the magnetic field with the magnetic flux. The role of magnetic helicity and other helicities (e.g. cross helicity) is currently not well understood in relation to MD and FD states, but these quantities are conjectured to be very important in the development of MD and FD states.
Bistability is also related to a very important phenomenon in dynamos - global field reversal. A strongly dipolar (MD) field can change to a transitional multipolar (FD) state before a reversal and then settle into another dipolar equilibrium (of opposite polarity) again after the reversal.This project aims to develop a coherent picture of how bistability operates in spherical dynamos. Since bistability is a fundamental property of dynamos, a characterisation of how bistable solutions form and develop is key for any deep understanding of planetary dynamos and, in particular, could be crucial for understanding magnetic field reversals.
Stellar atmospheres and their magnetic helicity fluxes (PhD)
Supervisors: Radostin Simitev, David MacTaggart, Robert Teed
Relevant research groups: Geophysical & Astrophysical Fluid Dynamics, Continuum Mechanics
Our Sun and many other stars have a strong large-scale magnetic field with a characteristic time variation. We know that this field is being generated via a dynamo mechanism driven by the turbulent convective motions inside the stars. The magnetic helicity, a quantifier of the field’s topology, is and essential ingredient in this process. In turbulent environments it is responsible for the inverse cascade that leads to the large-scale field, while the build up of its small-scale component can quench the dynamo.
In this project, the student will study the effects of magnetic helicity fluxes that happen below the stellar surface (photosphere), within the stellar atmosphere (chromosphere and corona) and between these two layers. This will be done using two-dimensional mean field simulations that allow parameter studies for different physical parameters. A fully three-dimensional model of a convective stellar wedge will then be used to provide a more detailed picture of the helicity fluxes and their effect on the dynamo. Using recent advancements that allow us to extract surface helicity fluxes from solar observations, the student will make use of observations to verify the simulation results. Other recent observational results on the stellar magnetic helicity will be used to benchmark the findings.
Efficient asymptotic-numerical methods for cardiac electrophysiology (PhD)
Supervisors: Radostin Simitev
Relevant research groups: Mathematical Biology, Continuum Mechanics
The mechanical activity of the heart is controlled by electrical impulses propagating regularly within the cardiac tissue during one's entire lifespan. A large number of very detailed ionic current models of cardiac electrical excitability are available.These realistic models are rather difficult for numerical simulations. This is due not only to their functional complexity but primarily to the significant stiffness of the equations.The goal of the proposed project is to develop fast and efficient numerical methods for solution of the equations of cardiac electrical excitation with the help and in the light of newly-developed methods for asymptotic analysis of the structure of cardiac equations (Simitev & Biktashev (2006) Biophys J; Biktashev et al. (2008), Bull Math Biol; Simitev & Biktashev (2011) Bull Math Biol)
The student will gain considerable experience with the theory of ordinary and partial differential equations, dynamical systems and bifurcation theory, asymptotic and perturbation methods,numerical methods. The applicant will also gain experience in computerprogramming, scientific computing and some statistical methods for comparison with experimental data.
Electrophysiological modelling of hearts with diseases (PhD)
Supervisors: Radostin Simitev
Relevant research groups: Mathematical Biology, Continuum Mechanics
The exact mechanisms by which heart failure occurs are poorly understood. On a more optimistic note, a revolution is underway in healthcare and medicine - numerical simulations are increasingly being used to help diagnose and treat heart disease and devise patient-specific therapies. This approach depends on three key enablers acting in accord. First, mathematical models describing the biophysical changes of biological tissue in disease must be formulated for any predictive computation to be possible at all. Second, statistical techniques for uncertainty quantification and parameter inference must be developed to link these models to patient-specific clinical measurements. Third, efficient numerical algorithms and codes need to be designed to ensure that the models can be simulated in real time so they can be used in the clinic for prediction and prevention. The goals of this project include designing more efficient algorithms for numerical simulation of the electrical behaviour of hearts with diseases on cell, tissue and on whole-organ levels. The most accurate tools we have, at present, are so called monolithic models where the differential equations describing constituent processes are assembled in a single large system and simultaneously solved. While accurate, the monolithic approaches are expensive as a huge disparity in spatial and temporal scales between relatively slow mechanical and much faster electrical processes exists and must be resolved. However, not all electrical behaviour is fast so the project will exploit advances in cardiac asymptotics to develop a reduced kinematic description of propagating electrical signals. These reduced models will be fully coupled to the original partial-differential equations for spatio-temporal evolution of the slow nonlinear dynamic fields. This will allow significantly larger spatial and time steps to be used in monolithic numerical schemes and pave the way for clinical applications, particularly coronary perfusion post infarction. The models thus developed will be applied to specific problems of interest, including (1) coupling among myocyte-fibroblast-collagen scar; (2) shape analysis of scar tissue and their effects on electric signal propagation; (3) personalized 3D heart models using human data. The project will require and will develop knowledge of mathematical modelling, asymptotic and numerical methods for PDEs and software development and some basic knowledge of physiology. Upon completion you will be a mature researcher with broad interdisciplinary education. You will not only be prepared for an independent scientific career but will be much sought after by both academia and industry for the rare combination of mathematical and numerical skills.
Fast-slow asymptotic analysis of cardiac excitation models (PhD)
Supervisors: Radostin Simitev
Relevant research groups: Mathematical Biology, Continuum Mechanics
Mathematical models of cardiac electrical excitation describe processess ocurring on a wide range of time and length scales.
Scalable approaches to mathematical modelling and uncertainty quantification in heterogeneous peatlands (PhD)
Supervisors: Raimondo Penta, Vinny Davies, Jessica Davies (Lancaster University), Lawrence Bull, Matteo Icardi (University of Nottingham)
Relevant research groups: Modelling in Space and Time, Environmental, Ecological Sciences & Sustainability, Machine Learning and AI, Emulation and Uncertainty Quantification, Continuum Mechanics
Funding: This project is competitively funded through the ExaGEO DLA.
While only covering 3% of the Earth’s surface, peatlands store >30% of terrestrial carbon and play a vital ecological role. Peatlands are, however, highly sensitive to climate change and human pressures, and therefore understanding and restoring them is crucial for climate action. Multiscale mathematical models can represent the complex microstructures and interactions that control peatland dynamics but are limited by their computational demands. GPU and Exascale computing advances offer a timely opportunity to unlock the potential benefits of mathematically-led peatland modelling approaches. By scaling these complex models to run on new architectures or by directly incorporating mathematical constraints into GPU-based deep learning approaches, scalable computing will to deliver transformative insights into peatland dynamics and their restoration, supporting global climate efforts.
Geophysical & Astrophysical Fluid Dynamics - Example Research Projects
Information about postgraduate research opportunities and how to apply can be found on the Postgraduate Research Study page. Below is a selection of projects that could be undertaken with our group.
Numerical simulations of planetary and stellar dynamos (PhD)
Supervisors: Radostin Simitev, Robert Teed
Relevant research groups: Geophysical & Astrophysical Fluid Dynamics, Continuum Mechanics
Using fluid dynamics and magnetohydrodynamics to model the magnetic fields of Earth, planets, the Sun and stars. Involves high-performance computing.
Observationally-constrained 3D convective spherical models of the solar dynamo (Solar MHD) (PhD)
Supervisors: Radostin Simitev, David MacTaggart, Robert Teed
Relevant research groups: Geophysical & Astrophysical Fluid Dynamics, Continuum Mechanics
Solar magnetic fields are produced by a dynamo process in the Solar convection zone by turbulent motions acting against Ohmic dissipation. Solar magnetic activity affects nearEarth space environment and can harm modern technology and endanger human health. Further, Solar magnetism poses fundamental physical and mathematical problems, e.g. about the nature of plasma turbulence and the topology of magnetic field generation. Current models of the global Solar dynamo fall in two classes (a) mean-field dynamos (b) convection-driven dynamos. The mean-field models are only phenomenological as they replace turbulent interactions by ad-hoc source and quenching terms. On the other hand, spherical convection-driven dynamo models are derived from basic principles with minimal assumptions and potentially offer true predictive power; these can also be extended to other stars and giant planets. However, at present, convection driven dynamo models operate in a wrong dynamical regime and have limited success in reproducing a number of important 1 observations including (a) the sunspot cycle period, polarity reversals and the sunspot butterfly diagram, (b) the poleward migration of diffuse surface magnetic fields, (c) the polar field strength and phase relationships between poloidal/toroidal components. The aims of this project are to (a) develop a three-dimensional convection-driven Solar dynamo model constrained by assimilation of helioseismic data, and (b) start to use the model to estimate turbulent properties that determine the internal dynamics and activity cycles of the Sun.
Modelling the force balance in planetary dynamos (PhD)
Supervisors: Robert Teed, Radostin Simitev
Relevant research groups: Geophysical & Astrophysical Fluid Dynamics, Continuum Mechanics
Current simulations of magnetic field (the 'dynamo process') generation in planets are run, not under the conditions of planetary cores and atmospheres, but in a regime idealised for computations. To forecast changes in planetary magnetic fields such as reversals and dynamo collapse, it is vital to understand the actual fluid dynamics of these regions. The aim of this project is to produce simulations of planetary cores and atmospheres with realistic force balances and, in doing so, understand how such force balances arise and affect the dynamics of the flow. The importance of different forces (e.g. Coriolis, Lorentz, viscous forces) determine the dynamics, the dynamo regime, and hence the morphology and strength of the magnetic field that is produced. This project would involve working with existing numerical code to perform the simulations and developing new techniques to determine the heirarchy of forces at play.
Identifying waves in dynamo models (PhD)
Supervisors: Robert Teed
Relevant research groups: Geophysical & Astrophysical Fluid Dynamics, Continuum Mechanics
This project would involve using existing (and developing new) techniques to isolate and study magnetohydrodynamic (MHD) waves in numerical calculations. Various classes of waves exist and may play a role in the dynamo process (which generates planetary magnetic fields) and/or help us better understand changes in the magnetic field.
Magnetic helicity as the key to dynamo bistability (PhD)
Supervisors: David MacTaggart, Radostin Simitev
Relevant research groups: Geophysical & Astrophysical Fluid Dynamics, Continuum Mechanics
The planets in the solar system exhibit very different types of large-scale magnetic field.The Earth has a strongly dipolar field, whereas the fields of other solar system planets, such as Uranus and Neptune, are far more irregular. Although the different physical compositions of the planets of the solar system will influence the behaviour of the large-scale magnetic fields that they generate, the morphology of planetary magnetic fields can depend on properties of dynamos common to all planets. Here, we refer to an important and recent discovery from dynamo simulations. Remarkably, two very different types of chaotic dipolar dynamo solutions have been found to exist over identical values of the basic parameters of a generic model of convection-driven dynamos in rotating spherical shells. The two solutions mentioned above can be characterised as ‘mean dynamos’, MD, where a strong poloidal field dominates and ‘fluctuating dynamos’, FD, where the poloidal component is weaker and the large-scale field can be described as multipolar. Although these two states have been shown to be bistable (co-exist) for a wide range of identical parameters, it is not clear how a particular state, MD or FD, is chosen and how/when one state can change to the other. Some of the bifurcations of such states has been investigated, but a deep understanding of the dynamics that cause the bifurcations remains to be developed. Since the magnetic topology of MD and FD states are fundamentally different, an important part of this project will be to probe the nature of MD and FD states by studying magnetic helicity, a magnetohydrodynamic invariant that combines information on the topology of the magnetic field with the magnetic flux. The role of magnetic helicity and other helicities (e.g. cross helicity) is currently not well understood in relation to MD and FD states, but these quantities are conjectured to be very important in the development of MD and FD states.
Bistability is also related to a very important phenomenon in dynamos - global field reversal. A strongly dipolar (MD) field can change to a transitional multipolar (FD) state before a reversal and then settle into another dipolar equilibrium (of opposite polarity) again after the reversal.This project aims to develop a coherent picture of how bistability operates in spherical dynamos. Since bistability is a fundamental property of dynamos, a characterisation of how bistable solutions form and develop is key for any deep understanding of planetary dynamos and, in particular, could be crucial for understanding magnetic field reversals.
Stellar atmospheres and their magnetic helicity fluxes (PhD)
Supervisors: Radostin Simitev, David MacTaggart, Robert Teed
Relevant research groups: Geophysical & Astrophysical Fluid Dynamics, Continuum Mechanics
Our Sun and many other stars have a strong large-scale magnetic field with a characteristic time variation. We know that this field is being generated via a dynamo mechanism driven by the turbulent convective motions inside the stars. The magnetic helicity, a quantifier of the field’s topology, is and essential ingredient in this process. In turbulent environments it is responsible for the inverse cascade that leads to the large-scale field, while the build up of its small-scale component can quench the dynamo.
In this project, the student will study the effects of magnetic helicity fluxes that happen below the stellar surface (photosphere), within the stellar atmosphere (chromosphere and corona) and between these two layers. This will be done using two-dimensional mean field simulations that allow parameter studies for different physical parameters. A fully three-dimensional model of a convective stellar wedge will then be used to provide a more detailed picture of the helicity fluxes and their effect on the dynamo. Using recent advancements that allow us to extract surface helicity fluxes from solar observations, the student will make use of observations to verify the simulation results. Other recent observational results on the stellar magnetic helicity will be used to benchmark the findings.
Integrable Systems and Mathematical Physics - Example Research Projects
Information about postgraduate research opportunities and how to apply can be found on the Postgraduate Research Study page. Below is a selection of projects that could be undertaken with our group.
There is also the possibility of applying to The EPSRC Centre for Doctoral Training in Algebra, Geometry, and Quantum Fields (AGQ CDT) is a collaborative effort by University of Glasgow, University of Edinburgh, and Heriot-Watt University. The programme is at the forefront of mathematical innovation, leveraging the power of symmetry, geometry, and quantum physics to shape the technologies of tomorrow.
Cherednik Algebras and related topics (PhD)
Supervisors: Misha Feigin
Relevant research groups: Algebra, Integrable Systems and Mathematical Physics
The project is aimed at clarifying certain questions related to Cherednik algebras. These questions may include study of homomorphisms between rational Cherednik algebras for particular Coxeter groups and special multiplicity parameters, defining and studying of new partial spherical Cherednik algebras and their representations related to quasi-invariant polynomials, study of differential operators on quasi-invariants related to non-Coxeter arrangements. Relations with quantum integrable systems of Calogero-Moser type may be explored as well. Some other possible topics may include study of quasi-invariants for non-Coxeter arrangements in relation to theory of free arrangements of hyperplanes.
Ultra-discrete Integrable Systems (PhD)
Supervisor: Claire Gilson
Relevant research groups: Integrable Systems and Mathematical Physics
An ultra-discrete equation is an equation where not only are the independent variables x and t discrete but the dependent variable are also discrete and often take on the values 0 or 1. There are many interesting results known about these systems. They are usually obtained by the process of ultra-discretization, this is a limiting process which takes you from a continuous integrable system to an ultra-discrete system via a discrete system. This project investigates the world of ultra-discrete systems and soliton cellular automata both periodic and non periodic. Looking in particular at how one can build different systems and extract the conserved quantities of these systems.
Non-commutative Integrable Systems (PhD)
Supervisor: Claire Gilson
Relevant research groups: Integrable Systems and Mathematical Physics
Integrable Systems are a very special class of differential equations with exact (soliton) solutions. There are several well known integrable systems, most of which are commutative (i.e the order in which the dependent variables are written down in the equation doesn't matter). Most of these equations have multiple soliton solutions that can be written down in terms of determinants such as Wronskians or Grammians. There are many fewer non-commutative integrable systems that are known, these have solutions in terms of quasi-determinants (Gelfand and Retakh, vol25, p91, Functional Analysis and its Applications). This project aims to investigate the properties of known and new non-commutative systems.
Quantum spin-chains and exactly solvable lattice models (PhD)
Supervisors: Christian Korff
Relevant research groups: Algebra, Integrable Systems and Mathematical Physics
Quantum spin-chains and 2-dimensional statistical lattice models, such as the Heisenberg spin-chain and the six and eight-vertex models remain an active area of research with many surprising connections to other areas of mathematics.
Some of the algebra underlying these models deals with quantum groups and Hecke algebras, the Temperley-Lieb algebra, the Virasoro algebra and Kac-Moody algebras. There are many unanswered questions ranging from very applied to more pure topics in representation theory and algebraic combinatorics. For example, recently these models have been applied in combinatorial representation theory to compute Gromov-Witten invariants (enumerative geometry) and fusion coefficients in conformal field theory (mathematical physics).
Integrable quantum field theory and Y-systems (PhD)
Supervisors: Christian Korff
Relevant research groups: Algebra, Integrable Systems and Mathematical Physics
The mathematically rigorous and exact construction of a quantum field theory remains a tantalising challenge. In 1+1 dimensions exact results can be found by computing the scattering matrices of such theories using a set of functional relations. These theories exhibit beautiful mathematical structures related to Weyl groups and Coxeter geometry.
In the thermodynamic limit (volume and particle number tend to infinity while the density is kept fixed) the set of functional relations satisfied by the scattering matrices leads to so-called Y-systems which appear in cluster algebras introduced by Fomin and Zelevinsky and the proof of dilogarithm identities in number theory.
Long-range quantum integrability (PhD)
Supervisors: Jules Lamers
Relevant research groups: Algebra, Integrable Systems and Mathematical Physics
This project deals with mathematical physics, or more precisely, representation theory with a view towards quantum physics. Spin chains are models for magnetic materials based on quantum mechanics. While research often assumes interactions between the spins of adjacent atoms only, long-range spin interactions are relevant for cutting-edge experiments, quantum information and computing, and theoretical high-energy physics.
If you like algebra or representation theory, and think you might want to help develop the powerful toolkit of quantum integrability in the long-range context, please send me an email! A background in physics is not required.
Keywords for possible topics are: Hecke algebras, Macdonald polynomials, quantum groups, elliptic generalisations, quantum spin chains, quantum many-body systems.
qDT invariants and deformations of hyperKahler geometry (PhD)
Supervisor: Ian Strachan
Relevant research groups: Geometry and Topology, Integrable Systems and Mathematical Physics
The project seeks to understand and exploit the integrable structure behind quantum Donaldson-Thomas invariants in terms of deformation of hyperKahler geometry and quantum-Riemann-Hilbert problems.
Almost-duality for arbitrary genus Hurwitz spaces (PhD)
Supervisor: Ian Strachan
Relevant research groups: Geometry and Topology, Integrable Systems and Mathematical Physics
The space of rational functions (interpreted as the space of holomorphic maps from the Riemann sphere to itself) may be endowed with the structure of a Frobenius manifolds, and hence there also exists an almost-dual Frobenius manifold structure. The class of examples include Coxeter and Extended-Affine-Weyl orbit group spaces. This extends to spaces of holomorphic maps between the torus and the sphere, where one can proved stronger results than just existence results. The project will seek to extend this to the explicit study of the space of holomorphic maps from an arbitrary genus Riemann surface to the Riemann sphere.
Mathematical Biology - Example Research Projects
Information about postgraduate research opportunities and how to apply can be found on the Postgraduate Research Study page. Below is a selection of projects that could be undertaken with our group.
There is also the possibility of applying to The Leverhulme Programme for Doctoral Training in Ecological Data Science which is hosted in our school. Information on how to apply can be found on the programme's application page.
Efficient asymptotic-numerical methods for cardiac electrophysiology (PhD)
Supervisors: Radostin Simitev
Relevant research groups: Mathematical Biology, Continuum Mechanics
The mechanical activity of the heart is controlled by electrical impulses propagating regularly within the cardiac tissue during one's entire lifespan. A large number of very detailed ionic current models of cardiac electrical excitability are available.These realistic models are rather difficult for numerical simulations. This is due not only to their functional complexity but primarily to the significant stiffness of the equations.The goal of the proposed project is to develop fast and efficient numerical methods for solution of the equations of cardiac electrical excitation with the help and in the light of newly-developed methods for asymptotic analysis of the structure of cardiac equations (Simitev & Biktashev (2006) Biophys J; Biktashev et al. (2008), Bull Math Biol; Simitev & Biktashev (2011) Bull Math Biol)
The student will gain considerable experience with the theory of ordinary and partial differential equations, dynamical systems and bifurcation theory, asymptotic and perturbation methods,numerical methods. The applicant will also gain experience in computerprogramming, scientific computing and some statistical methods for comparison with experimental data.
Electrophysiological modelling of hearts with diseases (PhD)
Supervisors: Radostin Simitev
Relevant research groups: Mathematical Biology, Continuum Mechanics
The exact mechanisms by which heart failure occurs are poorly understood. On a more optimistic note, a revolution is underway in healthcare and medicine - numerical simulations are increasingly being used to help diagnose and treat heart disease and devise patient-specific therapies. This approach depends on three key enablers acting in accord. First, mathematical models describing the biophysical changes of biological tissue in disease must be formulated for any predictive computation to be possible at all. Second, statistical techniques for uncertainty quantification and parameter inference must be developed to link these models to patient-specific clinical measurements. Third, efficient numerical algorithms and codes need to be designed to ensure that the models can be simulated in real time so they can be used in the clinic for prediction and prevention. The goals of this project include designing more efficient algorithms for numerical simulation of the electrical behaviour of hearts with diseases on cell, tissue and on whole-organ levels. The most accurate tools we have, at present, are so called monolithic models where the differential equations describing constituent processes are assembled in a single large system and simultaneously solved. While accurate, the monolithic approaches are expensive as a huge disparity in spatial and temporal scales between relatively slow mechanical and much faster electrical processes exists and must be resolved. However, not all electrical behaviour is fast so the project will exploit advances in cardiac asymptotics to develop a reduced kinematic description of propagating electrical signals. These reduced models will be fully coupled to the original partial-differential equations for spatio-temporal evolution of the slow nonlinear dynamic fields. This will allow significantly larger spatial and time steps to be used in monolithic numerical schemes and pave the way for clinical applications, particularly coronary perfusion post infarction. The models thus developed will be applied to specific problems of interest, including (1) coupling among myocyte-fibroblast-collagen scar; (2) shape analysis of scar tissue and their effects on electric signal propagation; (3) personalized 3D heart models using human data. The project will require and will develop knowledge of mathematical modelling, asymptotic and numerical methods for PDEs and software development and some basic knowledge of physiology. Upon completion you will be a mature researcher with broad interdisciplinary education. You will not only be prepared for an independent scientific career but will be much sought after by both academia and industry for the rare combination of mathematical and numerical skills.
Fast-slow asymptotic analysis of cardiac excitation models (PhD)
Supervisors: Radostin Simitev
Relevant research groups: Mathematical Biology, Continuum Mechanics
Mathematical models of cardiac electrical excitation describe processess ocurring on a wide range of time and length scales.
Mathematical models of vasculogenesis (PhD)
Supervisors: Peter Stewart
Relevant research groups: Mathematical Biology, Continuum Mechanics
Vasculogenesis is the process of forming new blood vessels from endothelial cells, which occurs during embryonic development. Viable blood vessels facilitate tissue perfusion, allowing the tissue to grow beyond the diffusion-limited size. However, in the absence of vasculogenesis, efforts to engineer functional tissues (eg for implantation) are restricted to this diffusion-limited size. This project will investigate mathematical models for vasculogenesis and explore mechanisms to stimulate blood vessel formation for in vitro tissues. The project will involve collaboration with Department of Biological Engineering at MIT, as part of the SofTMechMP project.
A coupled cardiovascular-respiration model for mechanical ventilation (PhD)
Supervisors: Peter Stewart, Nicholas A Hill
Relevant research groups: Mathematical Biology, Continuum Mechanics
Mechanical ventilation is a clinical treatment used to draw air into the lungs to facilitate breathing, used in treatment of premature babies with respiratory distress syndrome and in the treatment of severe Covid pneumonia. The aim is to oxygenate the blood while simultaneously removing unwanted by-products. However, over-inflation of the lungs can reduce the blood supply to the gas exchange surfaces, leading to a ventilation-perfusion mis-match. This PhD project will give you the opportunity to develop a mathematical model to describe the coupling between blood flow in the pulmonary circulation and air flow in the lungs (during both inspiration and expiration). You will devise a coupled computational framework, capable of testing patient-specific ventilation protocols. This is an ideal project for a postgraduate student with an interest in applying mathematical modelling and image analysis to predictive healthcare. The project will give you the opportunity to join a cross-disciplinary Research Hub that aims to push the boundaries of quantitative medicine and improve clinical decision making using innovative mathematical and statistical modelling.